Why Is Cos 120° Negative and How Is It Calculated?
The concept of value of cos 120 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Knowing the value of trigonometric functions at standard angles like 120 degrees is essential for quick calculations in school, competitive exams, and science subjects.
What Is Value of Cos 120?
The value of cos 120 refers to the cosine of a 120-degree angle. In trigonometry, cosine represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Cos 120 is especially important in topics like the unit circle, trigonometric ratios, and when working with angles in standard intervals. You’ll find this concept applied in areas such as geometry problems, vectors, and graphical representation of trigonometric functions.
Key Formula for Value of Cos 120
Here’s the standard formula: \( \cos(120^{\circ}) = \cos(180^{\circ} - 60^{\circ}) = -\cos(60^{\circ}) = -\frac{1}{2} \)
Cross-Disciplinary Usage
The value of cos 120 is not only useful in Maths but also plays an important role in Physics, Computer Science, and logical reasoning. When working with vectors, geometry, or even circuit analysis, cos 120 can appear in calculations. Students preparing for JEE, NEET, or board exams will see its relevance in various types of questions and word problems.
Step-by-Step Illustration
- Start with the standard angle: \( \cos(120^\circ) \)
- Express 120° as (180° – 60°):
\( \cos(120^\circ) = \cos(180^\circ - 60^\circ) \) - Use the trigonometric identity \( \cos(180^\circ-\theta) = -\cos\theta \):
\( \cos(120^\circ) = -\cos(60^\circ) \) - Recall that \( \cos(60^\circ) = \frac{1}{2} \):
So \( \cos(120^\circ) = -\frac{1}{2} \) - Final Answer: Cos 120° = -½ or -0.5
Standard Values Table for Cosine
| Angle (°) | Cosine Value |
|---|---|
| 0 | 1 |
| 30 | √3/2 |
| 45 | 1/√2 |
| 60 | 1/2 |
| 90 | 0 |
| 120 | -1/2 |
| 180 | -1 |
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for remembering the value of cos 120:
- If the angle is 120°, notice that it’s 60° past 60°, i.e., 120° = 180° – 60°.
- In the second quadrant (between 90° and 180°), cosine is negative.
- So, just take the positive cosine of 60° (which is ½) and put a minus sign.
- Answer: cos 120° = -½
Tricks like quadrant sign rules (All Students Take Calculus — CAST) help you quickly determine signs for all trigonometric values. Vedantu’s live and recorded sessions offer many such fast revision strategies for board and entrance exams.
Try These Yourself
- Calculate cos 150° using a similar method as cos 120°.
- Find the value of cos 240°.
- What is cos(180° – x) in terms of cos x?
- Is the value of cos 120° equal to the value of sin 30°? Explain why.
Frequent Errors and Misunderstandings
- Thinking the value of cos 120 is positive because 120 is less than 180 (remember: in the second quadrant, only sine is positive).
- Forgetting to use a negative sign for cosine between 90° and 180°.
- Confusing cos 120° with cos 60° — always check the quadrant and sign.
Relation to Other Concepts
The idea of value of cos 120 connects closely with topics such as trigonometric tables, identities, and complimentary angles. Mastering this helps you with triangle problems, vector calculations, and transformations in advanced classes.
Classroom Tip
A quick way to remember the value of cos 120 is to always visualize the unit circle and recall that in the second quadrant, cosine values are always negative. Vedantu’s teachers often draw the CAST diagram (All Students Take Calculus) to help students assign signs correctly and quickly solve trigonometric questions.
We explored the value of cos 120—from its definition and formula to its stepwise calculation, common mistakes, and how it ties in with other trigonometric ideas. Practice more trigonometry with Vedantu’s expert resources and interactive sessions to become confident at using these values in any context!
Internal Links to Related Topics
- Trigonometric Values of Standard Angles
- Unit Circle Explanation
- Trigonometric Identities
- Sin, Cos, Tan Values Table
FAQs on What is the Value of Cos 120° in Trigonometry?
1. What is the value of cos 120?
The value of cos 120° is $-\frac{1}{2}$. This is because 120° lies in the second quadrant where the cosine function is negative. Using the identity $\cos(180° - \theta) = -\cos \theta$, we get:
$$\cos 120° = \cos(180° - 60°) = -\cos 60° = -\frac{1}{2}$$
2. What is the value of cosec 120?
The value of cosec 120° is $\frac{2}{\sqrt{3}}$. Cosec is the reciprocal of the sine function. Since $\sin 120° = \frac{\sqrt{3}}{2}$,
$$\csc 120° = \frac{1}{\sin 120°} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}$$
3. What is the value of sin 120 in fraction?
The fractional value of sin 120° is $\frac{\sqrt{3}}{2}$. This value is derived as follows:
$$\sin 120° = \sin(180° - 60°) = \sin 60° = \frac{\sqrt{3}}{2}$$
4. Why is cos 120 cos 60?
This is related to the trigonometrical identity:
- $\cos(180° - \theta) = -\cos \theta$
5. How do you find the value of cos 120 degrees using the unit circle?
On the unit circle, the cosine of 120° is the x-coordinate of the corresponding point. Since 120° is in the second quadrant and forms a reference angle of 60°, the x-coordinate is negative:
- For 60°, $\cos 60° = \frac{1}{2}$
- For 120°, $\cos 120° = -\frac{1}{2}$
6. What is the relationship between cos 120° and cos 240°?
Cos 120° and cos 240° are related because both angles are equidistant from 180°.
- $\cos 120° = \cos(180° - 60°) = -\cos 60° = -\frac{1}{2}$
- $\cos 240° = \cos(180° + 60°) = -\cos 60° = -\frac{1}{2}$
7. Why is the cosine of 120° negative?
The cosine of 120° is negative because 120° lies in the second quadrant of the coordinate plane. In this quadrant, the x-values (cosine values) of the unit circle are negative. Therefore, $\cos 120° = -\frac{1}{2}$.
8. How do you calculate cos 120° without a calculator?
You can calculate cos 120° without a calculator by using trigonometric identities:
- Recognize that $120° = 180° - 60°$
- Apply the identity: $\cos(180° - \theta) = -\cos \theta$
- So, $\cos 120° = -\cos 60° = -\frac{1}{2}$
9. What is the general formula for cos (180° - θ)?
The general trigonometric identity for cos (180° - θ) is:
$$\cos(180° - \theta) = -\cos \theta$$
This relation is useful for finding the cosine of angles in the second quadrant.
10. How does Vedantu help students learn trigonometric values like cos 120°?
Vedantu provides comprehensive lessons, live classes, and practice resources to help students understand topics like trigonometric values.
- Detailed explanations and visualizations for concepts such as cos 120°
- Practice questions and solutions with step-by-step guidance
- Personalized tutoring for clarifying doubts related to trigonometric functions
